Methods of Math. Physics
Dr. E.J. Zita, The Evergreen State College
Lab II Rm 2272, zita@evergreen.edu
Winter wk 4, Thursday 27 Jan. 2011
• Griffiths Ch.2: Fields & Potentials
• Use Gauss’ Law in symmetric geometries
• Boundary conditions
• Energy stored in charge distributions
• Conductors and capacitors
E field due to infinite sheet of charge
+++++++++++++++++++++++++++++++++
uniform surface charge density s = q/area
s
E field due to (infinite) Parallel Plates
++++++++++++++++++
-------------------
s = dq/dA
-s
Parallel plates
Conservative fields admit potentials
 
 E  dl = 0
 
V = -  E  dl

 
 E = 0
b
Therefore
depends only on endpoints.
a
• Easy to find E from V
• V
is independent of choice of reference point V=0
• V is uniquely determined by boundary conditions
• Every central force (curl F = 0) is conservative (prob 2.25)
• Ex.2.5 p.74: parallel plates
Potential between Parallel plates
++++++++++++++++++
-------------------
s = dq/dA
-s
Draw the direction of the Electric field
Describe the potential (difference)
E = -V
V = -  E  dl
Units of potential
E = -V
V = -  E  dl
Capacitance is geometric
• Capacitance = Q/V = the amount of charge
that can be stored by a given potential
DIFFERENCE
• C depends on the geometry of the system
(and on the dielectric between the plates)
Parallel plate capacitor
Prob. 2.40 p.106: Energy between parallel plates
Ex. 2.10 p.104: Find the capacitance between two metal plates
of surface area A held a distance d apart.
Electrostatic boundary conditions:
• E is discontinuous across a charge layer: DE = s/e0
• E|| and V are continuous
• Prob 2.30 (a) p.90: check BC for parallel plates
Electrostatic potential energy
Shall we do a cylindrically
symmetric problem?
(Coax cable: 2.16 p.75, 2.24 p.82)
charges ↔ electric fields ↔
potentials
Properties of conductors:
(in electrostatics)
•
•
•
•
•
E=0 inside a conductor
r=0 inside a conductor
Any net charge will be found on the surface
A conductor is an equipotential
E is perpendicular to the surface, just
outside the conductor
Coming up next – Modern physics
H atom in wave mechanics